So when I just computed as normal way and I found out by software that the integral of $\sin\left(\,y - x \over x + y\right)$ is some crazy expression.
So I think there should be some other way to solve this question, is it something related to change of coordinate ? If yes, how?

Is for change of coordinates. Write $u=y-x$ and $v=x+y$. Then $y=\frac{1}{2}u+\frac{1}{2}v$ and $x=\frac{1}{2}v-\frac{1}{2}u$. Now, we must transform the region in the $xy$ plane in a region of the plane $uv$ as follow: Let be $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ the transform
$$T(x,y)=(y-x, x+y):=(u,v). $$
This transform send the trapezoidal region $S$ on $xy$ plan in the region in $uv$ plan with coordinates $(-1,1); (1,1); (2,2)$ and $(-2,2)$ (make a drawing).